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Full text of "Treatise On Analysis Vol-Ii"

6    PARTICULAR CASES OF CONVOLUTION OF MEASURES        273

(14.5.4) If the sequence (Mi)i^i^n & convolvable, and ifSt = Supp(/^) (13.19),
then Supp(^! * ti2 * * " * /O c Si$2    Srt , and the two sides are equal if all
the measures fa are positive.

Let x be a point not in the closure of S^    Srt, and let V be an open
neighborhood of x not meeting S1S2  Sw. If /is any continuous function
whose support is contained in V, we can write (13.21.18)

' ' '

=     ^iOi) ' ' 
Jsi                 Js

But if xt e Sf for 1 <; i ^ /i, we have f(xlx2    xn) = 0 by hypothesis, and the
first assertion of (14.5.4) is proved. If all the ^ are positive, then so is

If U is a 0-negligible open set, K a compact subset of U, and/e <2fR(G) a
function with values in [0, 1] which is equal to 1 on K and to 0 on (}U (4.5.2),
then we have

H

= 0;

therefore the open set of points (xl9..., xn) such that f(x1x2 '   xn) > % is
(//!  jU2  " *  /0-negligible, and consequently does not intersect the
support Sj x S2 x   x Sn of this measure (13.21.18), Since the mapping
(jcl9 ..., xn)\-+x1x2 *  -xn is continuous, we conclude that K does not intersect
SiS2 " " SB, and therefore that S^   * Sn is contained in Supp(ju). This com-
pletes the proof.

6. EXAMPLES AND PARTICULAR CASES OF CONVOLUTION
OF MEASURES

(14.6.1)   A Dirac measure es (13.1.3) on G is convolvable (on either side) with
every measure n on G, and we have

(14.6.1.1)                     BS * IJL = y(j)0,       M * es = 8(^~ 1)/^5

(14.6.1.2)                                               s* et = est.ft with v, or